Properties

Label 2-8032-1.1-c1-0-131
Degree $2$
Conductor $8032$
Sign $1$
Analytic cond. $64.1358$
Root an. cond. $8.00848$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.00·3-s + 2.74·5-s − 3.93·7-s + 6.00·9-s + 2.93·11-s − 2.73·13-s + 8.25·15-s + 0.166·17-s + 0.325·19-s − 11.8·21-s − 0.107·23-s + 2.56·25-s + 9.00·27-s + 7.02·29-s − 4.61·31-s + 8.80·33-s − 10.8·35-s − 4.53·37-s − 8.19·39-s − 5.07·41-s + 12.0·43-s + 16.5·45-s + 5.57·47-s + 8.47·49-s + 0.500·51-s + 7.42·53-s + 8.06·55-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.22·5-s − 1.48·7-s + 2.00·9-s + 0.884·11-s − 0.757·13-s + 2.13·15-s + 0.0404·17-s + 0.0747·19-s − 2.57·21-s − 0.0224·23-s + 0.512·25-s + 1.73·27-s + 1.30·29-s − 0.829·31-s + 1.53·33-s − 1.82·35-s − 0.746·37-s − 1.31·39-s − 0.791·41-s + 1.83·43-s + 2.46·45-s + 0.812·47-s + 1.21·49-s + 0.0701·51-s + 1.01·53-s + 1.08·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8032\)    =    \(2^{5} \cdot 251\)
Sign: $1$
Analytic conductor: \(64.1358\)
Root analytic conductor: \(8.00848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8032,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.701865936\)
\(L(\frac12)\) \(\approx\) \(4.701865936\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 + T \)
good3 \( 1 - 3.00T + 3T^{2} \)
5 \( 1 - 2.74T + 5T^{2} \)
7 \( 1 + 3.93T + 7T^{2} \)
11 \( 1 - 2.93T + 11T^{2} \)
13 \( 1 + 2.73T + 13T^{2} \)
17 \( 1 - 0.166T + 17T^{2} \)
19 \( 1 - 0.325T + 19T^{2} \)
23 \( 1 + 0.107T + 23T^{2} \)
29 \( 1 - 7.02T + 29T^{2} \)
31 \( 1 + 4.61T + 31T^{2} \)
37 \( 1 + 4.53T + 37T^{2} \)
41 \( 1 + 5.07T + 41T^{2} \)
43 \( 1 - 12.0T + 43T^{2} \)
47 \( 1 - 5.57T + 47T^{2} \)
53 \( 1 - 7.42T + 53T^{2} \)
59 \( 1 - 13.1T + 59T^{2} \)
61 \( 1 + 1.26T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 - 2.67T + 73T^{2} \)
79 \( 1 - 7.51T + 79T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 - 6.98T + 89T^{2} \)
97 \( 1 - 2.36T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.894374332599659696044782219659, −7.01879472234922920209923551247, −6.69151522161751317684672041569, −5.89741641180516307330169370715, −5.00111421528144537785709534459, −3.84694248017402544119533198437, −3.50467366225167718177035867939, −2.42437658541514324985880725965, −2.28272474108637848869499773990, −1.00430442033054419815484532660, 1.00430442033054419815484532660, 2.28272474108637848869499773990, 2.42437658541514324985880725965, 3.50467366225167718177035867939, 3.84694248017402544119533198437, 5.00111421528144537785709534459, 5.89741641180516307330169370715, 6.69151522161751317684672041569, 7.01879472234922920209923551247, 7.894374332599659696044782219659

Graph of the $Z$-function along the critical line